Hybrid classical-quantum algorithm for solving the incompressible Navier-Stokes equations on quantum hardware

Efficient PDE solvers for Navier-Stokes-like problems are critical to solving many science and engineering problems. Quantum algorithms have been proposed to solve PDEs and verified on small-scale simulators. In the (very) long term, PDEs can be solved via method-of-lines-like approaches and a Harrow-Hassidim-Lloyd (HHL) algorithm, bringing an exponential speedup over classical methods. For near-term quantum hardware (NISQ-era), variational quantum algorithms such as the Variational Quantum Eigensolver (VQE) are more appropriate, entailing fewer quantum gates and are thus more robust to noise. Here, we solve the incompressible Navier-Stokes equation via a classical-hybrid solution strategy. Variational methods are used to solve the Poisson equation and enforce incompressibility, while nonlinear inertial effects inappropriate for a quantum algorithm are computed classically. Both methods leverage the advantages of their respective compute substrates and exchange requisite information at each time step. Benchmarks for canonical flow problems on simulators and quantum hardware determine the required and realized solution fidelity on noisy quantum hardware.